Independence clustering (without a matrix)
This addresses a fundamental clustering challenge for scenarios where mutual independence is key, such as in time series analysis, but it appears incremental as it builds on existing independence concepts with new algorithmic approaches.
The paper tackles the independence clustering problem, where the goal is to partition a set of random variables into the finest clusters such that clusters are mutually independent, using only sample data without pairwise similarity measurements. It proposes consistent, computationally tractable algorithms for i.i.d. and stationary time series settings, with a focus on the latter.
The independence clustering problem is considered in the following formulation: given a set $S$ of random variables, it is required to find the finest partitioning $\{U_1,\dots,U_k\}$ of $S$ into clusters such that the clusters $U_1,\dots,U_k$ are mutually independent. Since mutual independence is the target, pairwise similarity measurements are of no use, and thus traditional clustering algorithms are inapplicable. The distribution of the random variables in $S$ is, in general, unknown, but a sample is available. Thus, the problem is cast in terms of time series. Two forms of sampling are considered: i.i.d.\ and stationary time series, with the main emphasis being on the latter, more general, case. A consistent, computationally tractable algorithm for each of the settings is proposed, and a number of open directions for further research are outlined.